Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity

Abstract

In solving the split variational inequality problems, very few methods have been considered in the literature and most of these few methods require the underlying operators to be co-coercive. This restrictive co-coercive assumption has been dispensed with in some methods, many of which require a product space formulation of the problem. However, it has been discovered that this product space formulation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting structure of the split variational inequality problem. In this paper, we present two new methods with inertial steps for solving the split variational inequality problems in real Hilbert spaces without any product space formulation. We prove that the sequence generated by these methods converges strongly to a minimum-norm solution of the problem when the operators are pseudomonotone and Lipschitz continuous. Also, we provide several numerical experiments of the proposed methods in comparison with other related methods in the literature.

Appendices

Appendix 1: Algorithm 1 of He et al. [24]

Stop 0. :

Given a symmetric positive definite matrix $\mathbb {{H}}\in \mathbb {R}^{m\times m},~~\gamma \in (0,2)$ and $\rho \in (\rho _{\min \limits },~3)$, where $\rho _{\min \limits } := \max \limits \left \{-3, \frac {2(v-1)\mu _{1}}{4\zeta _{\max \limits }(T'{H}T)}\right \}$ and $T:\mathbb {R}^{N}\to \mathbb {R}^{m}$ is a linear operator (where $T^{\prime }$ means the transpose of T). Set an initial point $\textbf {u}_{1}:=(x_{1},y_{1},\lambda _{1})\in {\Omega }:=\mathcal {C}\times \mathcal {Q}\times \mathbb {R}^{m}$, where $\mathcal {C}$ and $\mathcal {Q}$ are nonempty closed and convex subsets of $\mathbb {R}^{N}$ and $\mathbb {R}^{m}$, respectively.

Step 1. :

Generate a predictor $\tilde {\textbf {u}}_{n}:=(\tilde {x}_{n},\tilde {y}_{n},\tilde {\lambda }_{n})$ with appropriate parameters μ₁ and μ₂ :

$$ {} \left\{ \begin{array}{ll} \bar{\lambda}_{n}=\lambda_{n}-\mathbb{H}(Tx_{n}-y_{n}),\\ \tilde{x}_{n}=P_{\mathcal{C}}\left[x_{n}-\frac{1}{\mu_{1}}\left( A(x_{n})-T^{\prime}\bar{\lambda}_{n}\right)\right],\\ \hat{\lambda}_{n}=\lambda_{n}-\mathbb{H}(T\hat{x}_{n}-y_{n}),~~~\text{where}~~~ \hat{x}_{n}:=\rho x_{n}+(1-\rho)\tilde{x}_{n},\\ \tilde{y}_{n}=P_{\mathcal{Q}}\left[y_{n}-\frac{1}{\mu_{2}}(F(y_{n})+\hat{\lambda}_{n})\right],\\ \tilde{\lambda}_{n}=\lambda_{n}-\mathbb{H}(T\tilde{x}_{n}-\tilde{y}_{n}). \end{array} \right. $$

(1)

Step 2. :

Update the next iterative u_n+ 1 := (x_n+ 1,y_n+ 1,λ_n+ 1) via

$$ \textbf{u}_{n+1}:=\textbf{u}_{n}-\gamma \alpha_{k} d(\textbf{u}_{n},\tilde{\textbf{u}}_{n}), $$

where

$$ \left\{ \begin{array}{ll} \alpha_{k}:=\frac{\psi(\textbf{u}_{n},\tilde{\textbf{u}}_{n})}{\|d(\textbf{u}_{n},\tilde{\textbf{u}}_{n})\|^{2}},\\ d({\textbf{u}}_{n},\tilde{\textbf{u}}_{n}):=G(\textbf{u}_{n}-\tilde{\textbf{u}}_{n})-\xi_{n},\\ \psi(\textbf{u}_{n},\tilde{\textbf{u}}_{n}):=\left\langle \lambda_{n}-\tilde{\lambda}_{n}, \rho T(x_{n}-\tilde{x}_{n})-(y_{n}-\tilde{y}_{n})\right\rangle+\left\langle \textbf{u}_{n}- \tilde{\textbf{u}}_{n}, d(\textbf{u}_{n}, \tilde{\textbf{u}}_{n})\right\rangle, \end{array} \right. $$

$\xi _{n}:= \begin {pmatrix} \xi _{n{_{x}}}\\~~~ \xi _{n{_{y}}}\\0\end {pmatrix}:=\begin {pmatrix} A(x_{n})-A(\tilde {x}_{n})+T^{\prime } \mathbb {H}T(x_{n}-\tilde {x}_{n})\\ F(y_{n})-F(\tilde {y}_{n})+\mathbb {H}(y_{n}-\tilde {y}_{n})\\0 \end {pmatrix},$ A and F are monotone and Lipschitz continuous with constants L₁ and L₂ respectively, and

$G:=\begin {pmatrix} \mu _{1}I_{N}+\rho T^{\prime }\mathbb {H}T&0&0\\ 0&\mu _{2}I_{m}+\mathbb {H}&0\\ 0&0& \mathbb {H}^{-1}\\ \end {pmatrix}$ is the block diagonal matrix, with identity matrices I_N and I_m of size N and m, respectively. The parameters μ₁ and μ₂ are chosen such that

$$ \begin{array}{@{}rcl@{}} \|\xi_{n_{x}}\|\leq v\mu_{1}\|x_{n}-\tilde{x}_{n}\|~~~~~\text{and}~~~~~~~~ \|\xi_{n_{y}}\|\leq v\mu_{2}~\|y_{k}-\tilde{y}_{n}\|,~\text{where}~ v\in(0,1). \end{array} $$

Appendix 2: The algorithm in Reich and Tuyen [44, Theorem 4.4]

For any initial guess $x_{1}=x\in {\mathscr{H}}_{1}$, define the sequence {x_n} by

$$ \begin{array}{@{}rcl@{}} y_{n}&=&VI(\mathcal{C}, ~\lambda_{n} A+I_{\mathcal{H}_{1}}-x_{n}),\\ z_{n}&=&VI(\mathcal{Q}, ~\mu_{n} f+I_{\mathcal{H}_{2}}-Ty_{n}),\\ \mathcal{C}_{n} &=&\{z\in \mathcal{H}_{1} : ||y_{n}-z||\leq ||x_{n}-z||\},\\ \mathcal{D}_{n}&=&\{z\in \mathcal{H}_{1} : ||z_{n}-Tz||\leq ||Ty_{n}-Tz||\},\\ \mathcal{W}_{n}&=&\{z\in \mathcal{H}_{1} : \langle z-x_{n}, x_{1}-x_{n} \rangle \leq 0\},\\ x_{n+1}&=&P_{\mathcal{C}_{n}\cap \mathcal{D}_{n}\cap \mathcal{W}_{n}}(x_{1}),~n\geq 1, \end{array} $$

where $I_{{\mathscr{H}}_{1}}$ and $I_{{\mathscr{H}}_{2}}$ are identity operators in ${\mathscr{H}}_{1}$ and ${\mathscr{H}}_{2}$, respectively, and {λ_n} and {μ_n} are two given sequences of positive numbers satisfying the following condition:

$$\min\left\lbrace \underset{n}{\inf} \{\lambda_{n}\},~~ \underset{n}{\inf} \{\mu_{n}\} \right\rbrace \geq r>0.$$

Appendix 3: Algorithm 1 of Pham et al. [41]

Step 0. :

Choose $\mu _{0}, \lambda _{0}>0, ~~\mu , \lambda \in (0, 1),~\{\tau _{n}\}\subset [\underline {\tau },~ \bar {\tau }]\subset \left (0, \frac {1}{||T||^{2}+1}\right ), ~\{\alpha _{n}\} \subset (0, 1)$ such that $\lim \limits _{n\to \infty } \alpha _{n}=0$ and ${\sum }_{n=1}^{\infty } \alpha _{n}=\infty .$

Step 1. :

Let $x_{1}\in {\mathscr{H}}_{1}$. Set n = 1.

Step 2. :

Compute

$$u_{n}=Tx_{n},$$

$$v_{n}=P_{\mathcal{Q}}(u_{n}-\mu_{n} fu_{n}),$$

$$w_{n}=P_{\mathcal{Q}_{n}}(u_{n}-\mu_{n} fv_{n}),$$

where

$$\mathcal{Q}_{n}=\{w_{2}\in \mathcal{H}_{2} : \langle u_{n}-\mu_{n} f u_{n}-v_{n}, w_{2}-v_{n}\rangle\leq 0\}$$

and

$$ \mu_{n+1}=\begin{cases} \min\left\lbrace\frac{\mu||u_{n}-v_{n}||}{||fu_{n}-fv_{n}||},~\mu_{n}\right\rbrace,& \text{if}~fu_{n}\neq fv_{n},\\ \mu_{n},& \text{otherwise}. \end{cases} $$

Step 3. :

Compute

$$y_{n}=x_{n}+\tau_{n} T^{*}(w_{n}-u_{n}),$$

$$z_{n}=P_{\mathcal{C}}(y_{n}-\lambda_{n} Ay_{n}),$$

$$t_{n}=P_{\mathcal{C}_{n}}(y_{n}-\lambda_{n} Az_{n}),$$

where

$$\mathcal{C}_{n}=\{w_{1}\in \mathcal{H}_{1} : \langle y_{n}-\lambda_{n} A y_{n}-z_{n}, w_{1}-z_{n}\rangle\leq 0\}$$

and

$$ \lambda_{n+1}=\begin{cases} \min\left\lbrace\frac{\lambda||y_{n}-z_{n}||}{||Ay_{n}-Az_{n}||},~\lambda_{n}\right\rbrace,& \text{if}~Ay_{n}\neq Az_{n},\\ \lambda_{n},& \text{otherwise}. \end{cases} $$

Step 4. :

Compute

$$x_{n+1}=\alpha_{n} x_{1}+(1-\alpha_{n}) t_{n}.$$

Set n := n + 1 and go back to Step 2.